During inelastic collision between two bodies, which of the following quantities always remain conserved? (a)

NCERT Exemplar · Q6

Q. During inelastic collision between two bodies, which of the following quantities always remain conserved?

(a) Total kinetic energy.

(b) Total mechanical energy.

(d) Speed of each body.

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Concept understanding — Conservation of Momentum in Collisions

Conservation of Momentum in Collisions

When two objects collide — two billiard balls, two train carriages coupling together, a bat striking a ball — each one exerts a force on the other for the brief instant they are in contact. This concept looks specifically at what those internal, mutual forces do to the total momentum of the colliding pair.

Setting Up the Idea

Consider two bodies of masses $m_1$ and $m_2$ , moving with velocities $\vec{u}_1$ and $\vec{u}_2$ just before they collide, and velocities $\vec{v}_1$ and $\vec{v}_2$ just after. During the collision, body 1 pushes on body 2 with some force $\vec{F}$ , and by Newton's third law, body 2 pushes back on body 1 with force $-\vec{F}$ — equal in size, opposite in direction, acting for exactly the same contact time $\Delta t$ .

Since impulse equals change in momentum ( $\vec{F}\Delta t = \Delta \vec{p}$ ):

Change in momentum of body 1: $-\vec{F}\Delta t$
Change in momentum of body 2: $+\vec{F}\Delta t$

Add the two changes together and they cancel to zero. Whatever momentum one body loses, the other gains — the total is untouched.

Important

For any collision where no external force acts on the pair (an isolated system), the total linear momentum just before the collision equals the total linear momentum just after:

$m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_2 \vec{v}_2$

Why "Internal" Forces Don't Change the Total

The collision force between the two bodies is an internal force of the two-body system — it acts between the objects, not from outside. Internal forces always come in Newton's-third-law pairs, and any equal-and-opposite pair produces zero net impulse on the system as a whole. External forces — gravity, friction from the floor, air resistance — are usually negligible during a collision because contact lasts only a fraction of a second, far too short for a small external force to produce a noticeable impulse.

Watch out

Momentum conservation applies to the system's total, not to each object individually. One ball can slow down drastically and the other speed up — their individual momenta change freely; only the sum stays fixed.

Worked Example

A 2 kg ball moving at 3 m/s (rightward) strikes a stationary 1 kg ball. After the collision the 2 kg ball moves at 1 m/s in the same direction. Find the velocity of the 1 kg ball.

Take rightward as positive.

p_{\text{before}} = (2)(3) + (1)(0) = 6\ \text{kg m/s}

Let $v$ be the final velocity of the 1 kg ball:

p_{\text{after}} = (2)(1) + (1)(v) = 2 + v

Setting these equal: $2 + v = 6 \implies v = 4\ \text{m/s}$ . …

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In any collision, total linear momentum is always conserved because no external force acts on the system — the correct option is (C).

The key to this question is understanding what "conserved" means in physics: a quantity is conserved if its total value for the system remains unchanged before and after the event. In a collision, we're dealing with two bodies interacting over a very short time, and the forces between them are internal to the system. That's the foundation.

Now, let's walk through each option.

Total kinetic energy — This is the classic trap. In an inelastic collision, kinetic energy is not conserved. Some of it gets converted into other forms: heat, sound, or permanent deformation of the bodies. The problem explicitly says "inelastic collision," so option (A) is out.
Total mechanical energy — Mechanical energy is the sum of kinetic and potential energy. In a typical collision on a horizontal surface, there's no change in gravitational potential energy, so mechanical energy would just be kinetic energy — and we already know that's not conserved. Even if the bodies are at different heights, the deformation during an inelastic collision dissipates energy as heat and sound, which is not mechanical energy. So (B) is also wrong.
Total linear momentum — This is the winner. Newton's third law tells us that the forces two bodies exert on each other are equal and opposite. Over the same time interval, the impulses are equal and opposite, so the change in momentum of one body exactly cancels the change in momentum of the other. As long as no external force acts on the system (like friction or gravity from outside), total momentum stays constant. In a collision, we always assume external forces are negligible compared to the huge internal impact forces. So momentum is conserved in every collision — elastic or inelastic. …

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Q.During inelastic collision between two bodies, which of the following quantities always remain conserved?

Concept understanding — Conservation of Momentum in Collisions

Conservation of Momentum in Collisions

Setting Up the Idea

Why "Internal" Forces Don't Change the Total

Worked Example

Concept: Momentum Conservation Is Universal for an Isolated System

Q.During inelastic collision between two bodies, which of the following quantities always remain conserved?

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